Question 778·Hard·Math

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If $2^{x/2} + 2^{x/3} = 12$ , what is the value of $x$ ?

Your answer

In an exponential equation, the unknown sits in an exponent. Here the powers are added, so you can't combine their exponents across the plus sign. Use a one-variable Desmos regression to find the input that makes the sum equal the target. Both terms grow as the input grows, so the equation can have only one solution.

Hints

Hint 1
The powers are added, not multiplied. The rule $2^a\cdot2^b=2^{a+b}$ doesn't work for $2^a+2^b$ . What happens to each term as $x$ gets larger?
Hint 2
Because the base $2$ is greater than $1$ , each exponential term grows as its exponent grows. Both exponents grow with $x$ , so their sum grows too. That means there is at most one solution.
Hint 3
For a one-unknown regression, replace every $x$ with $x_1$ and the equality sign with $\sim$ . Desmos then finds the value of $x_1$ under PARAMETERS.

Step-by-step

Solve the increasing sum with Desmos
Step 1
See why there can be only one solution
Both $x/2$ and $x/3$ increase as $x$ increases. Raising $2$ to a larger exponent gives a larger value, so both terms increase, and their sum can equal $12$ only once. Don't add the exponents: exponent rules combine powers that are multiplied, not powers that are added.
Step 2
Find the input that makes the sum 12
Type $2^{x_1/2}+2^{x_1/3}\sim12$ . The $x_1$ tells Desmos to solve for a number rather than graph $x$ , and $\sim$ makes this a regression. Under PARAMETERS, Desmos shows $x_1=6$ . So the value of $x$ is $6$ . Grid in 6.

Question 778·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this